Due to our new schedule, I only taught two classes today – Probability/Statistics and Calculus. In Calculus we are actually studying some elementary counting and probability ideas. For about 25 minutes today, we explored the birthday paradox. If you don’t know what that is, you can view it on this TED-Ed video. I showed the first 30 seconds or so of this video and paused it when it got to the question.
The main question being “How many people do you need to have in a room for there to be more than a 50% chance that at least two people share the same birthday?” This problem incorporates complementary events and independent events.
I started by having each of the 16 students write two dates on a single slip of paper. Their birthday and the birthday of someone they know. This doesn’t quite match the scenario, because it is very unlikely for a student to have his or her two listed dates be the same. However, in reality two people standing beside each other could actually share the same birthday. We did this twice and both times we found a match, which students found surprising.
Next, we used technology (TI-84) to randomly generate a list of 32 integers from 1 to 365. We stored these numbers in a list and then sorted the list to make it easier to identify if we had a match. Unfortunately, only 10 people had calculators on them, but we had 80% have a match the first time we used the technology. We had 80% with a match on our second trial and 100% match on our third trial.
To help students understand why this probability was so high, I played the remaining 4 minutes of the TED-Ed video. So, how many people do you need to have in a room for there to be more than a 50% chance that at least two people share the same birthday? The answer is about 23.